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Let $k$ be a positive real number. The square with vertices  (k, 0), (0, k), (-k, 0), and (0, -k) is plotted in the coordinate plane.
Find conditions on a > 0 and b > 0 such that the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ is contained inside the square (and tangent to all of its sides). HINTS: Suppose that the line x+y=k is tangent to the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1.$. Algebraically, what can we say about the solutions? In particular, the number of solutions?

May 22, 2020

#1
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What's an ellipse?

May 22, 2020
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Let $$k$$ be a positive real number. The square with vertices  $$(k, 0)$$, $$(0, k)$$, $$(-k, 0)$$, and $$(0, -k)$$ is plotted in the coordinate plane.
Find conditions on $$a > 0$$ and $$b > 0$$ such that the ellipse $$\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$$ is contained inside the square (and tangent to all of its sides).

I Suppose that the line $$\mathbf{y=x+ k}$$ is tangent to the ellipse $$\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$$.

May 22, 2020
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@heureka, what do $x_t$ and $y_t$ mean in your solution? Are they bases from logs?

Guest May 22, 2020
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@heureka, what do $x_t$ and $y_t$ mean in your solution? Are they bases from logs?

no

$$(x_t,y_t)$$ are the coordinates of the tanget point P at the ellipse on my line $$y = x+k$$. See my link and the graph.

The tanget point of the circle is $$(-10,10)$$, so $$x_t=-10$$ and $$y_t =10$$.

heureka  May 22, 2020
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@heureka, what does $1= -\frac {b^2}{a^2} \frac{x_t}{y_t}$ mean in your solution? Can you explain to me what that means?

Guest May 23, 2020